Supplementary Material to
“Discriminatory Nonlinear Pricing, Fixed Costs, and
Welfare in Intermediate-Goods Markets”
Fabian Herwega , Daniel Müllerb
a
b
University of Bayreuth
University of Würzburg
Appendix B. Supplementary Material to “General Nonlinear Tariffs” (Proof of Proposition 4)
Instead of two-part tariffs, the manufacturer now offers menus of quantitytransfer pairs to the downstream firms. Here, downstream firms acquire a
given quantity at a pre-specified price, such that they indeed might have
an incentive not to convert all units of the input into the final product. In
this regard, the assumption of free disposal gives leeway in decision-making
to the downstream firms and weakens the position of the manufacturer. In
what follows, we initially abstract from free disposal of the input good; i.e.,
we consider quantity forcing contracts, under which a downstream firm sells
the same amount of the final consumption good as it acquired from the input
good. In the end, we will show that—under the additional assumption that
K > k—the allocation implemented by the optimal quantity forcing contract
also prevails under free disposal, such that this contract must also be optimal
if downstream firms can freely dispose of the input.
Slightly abusing the notation introduced in the paper, let
π(q, ki ) = q[P (q) − ki ]
(B.1)
denote the “gross” profits (i.e., profits before subtracting any transfer payment and the fixed cost) of a firm from selling quantity q when operating at
marginal cost ki ∈ {0, k}.
Price Discrimination.—If price discrimination is allowed, the manufacturer can offer each retailer an individualized contract and therefore, effecPreprint submitted to International Journal of Industrial Organization
April 4, 2016
tively, solves two independent optimization problems. Consider the manufacture’s contract offer to downstream firm i ∈ {0, k} that operates at marginal
cost ki , where k0 = 0 < k = kk , and associated fixed cost Fi . The logic of the
revelation principle implies that maximum upstream profits can be achieved
by an individually rational contract of the form {(qi , ti )}. In consequence,
the manufacturer solves the following problem:
max ti − Kqi
(qi ,ti )
subject to
(PCi ) π(qi , ki ) − ti ≥ Fi
Under the optimal contract, the (PCi ) constraint has to bind. In consequence, the manufacturer effectively chooses quantity qi to maximize the
profits of a vertically integrated firm, which leads to the following observation:
Lemma 3. Let qid denote the optimal quantity offered to retailer i ∈ {0, k}
under price discrimination. Then, qid = q JS (ki ).
Uniform Pricing.—If price discrimination is banned, the manufacturer
has to offer one and the same contract to both downstream firms. As there
are two types of downstream firms, the logic of the revelation principle implies
that maximum upstream profits can be achieved by an incentive compatible
and individually rational menu of the form {(q0 , t0 ), (qk , tk )}, where (qi , ti )
is the quantity-transfer pair designated to downstream firm i ∈ {0, k}. In
consequence, the manufacturer solves the following problem:
max
(q0 ,t0 ),(qk ,tk )
[t0 − Kq0 ] + [tk − Kqk ]
subject to
(PC0 ) π(q0 , 0) − t0 ≥ F0
(PCk ) π(qk , k) − tk ≥ Fk
(IC0 ) π(q0 , 0) − t0 ≥ π(qk , 0) − tk
(ICk ) π(qk , k) − tk ≥ π(q0 , k) − t0
We start with some basic, yet important, observations. First, both (IC0 ) and
(ICk ) being simultaneously satisfied implies that the following monotonicity
requirement is satisfied:
qk ≤ q0 .
(MON)
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Second, under the optimal contract, either (PCi ) or (ICi ) (or both) has to be
binding. Third, if the monotonicity requirement is satisfied and one retailer’s
incentive compatibility constraint binds, then the other retailer’s incentive
compatibility constraint is automatically satisfied. Hence, under the optimal
contract, at most one incentive compatibility constraint imposes a binding
restriction.
Which constraints actually impose a binding restriction is determined by
the ratio of the difference in fixed costs to the difference in marginal costs,
F
with F := F0 − Fk . We next present a detailed analysis of the cases that
k
have to be distinguished.
Case I: (IC0 ) and (PCk ) bind.
If (IC0 ) and (PCk ) are binding, transfers are given by
t0 = π(q0 , 0) − kqk − Fk
and
tk = π(qk , k) − Fk .
(B.2)
Ignoring (ICk ) and (PC0 ) for the moment, the manufacturer’s problem amounts
to
max [π(q0 , 0) − Kq0 ] + [π(qk , k) − (K + k)qk ] − 2Fk
q0 ,qk
(B.3)
From the manufacturer’s objective function it becomes apparent that the
optimal quantity to offer the retailer with low marginal cost is q0I = q JS (0).
Differentiation w.r.t. qk reveals that the optimal quantity to offer the retailer
with high marginal cost is qkI = 0 if P (0) ≤ 2k + K. If P (0) > 2k + K, on the
other hand, the retailer with high marginal cost is offered a strictly positive
quantity qkI > 0, which is implicitly characterized by
P (qkI ) + qkI P 0 (qkI ) = K + 2k.
(B.4)
It remains to check whether the neglected constraints are satisfied. By assumption the left-hand side of (B.4) is decreasing in q, such that qkI < q JS (k).
This, in turn, implies that the monotonicity requirement (MON) is satisfied.
As (IC0 ) binds, (ICk ) then is automatically satisfied. Finally, (PC0 ) is satisfied as long as
F0 ≤ kqkI + Fk ⇐⇒ qkI ≥
F
.
k
(B.5)
Case II: (IC0 ), (PC0 ), and (PCk ) bind.
If (PC0 ) and (PCk ) are binding, transfers are given by
t0 = π(q0 , 0) − F0
tk = π(qk , k) − Fk .
and
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(B.6)
Inserting these transfers into the binding (IC0 ) constraint pins down the
quantity optimally offered to the retailer with high marginal cost:
F0 = π(qk , 0) − π(qk , k) + Fk ⇒ qkII =
F0 − Fk
.
k
(B.7)
Ignoring (ICk ) for the moment, the manufacturer’s problem amounts to
max [π(q0 , 0) − Kq0 ] + [π(qkII , k) − KqkII ] − F0 − Fk .
q0
(B.8)
Hence, the optimal quantity to offer the retailer with low marginal cost is
q0II = q JS (0).
The monotonicity requirement (MON) is satisfied as long as
q JS (0) ≥
F
,
k
(B.9)
in which case also (ICk ) is satisfied because (IC0 ) binds.
Case III: (PC0 ) and (PCk ) bind.
If (PC0 ) and (PCk ) are binding, transfers are given by
t0 = π(q0 , 0) − F0
tk = π(qk , k) − Fk .
and
(B.10)
Ignoring (IC0 ) and (IC0 ) for the moment, the manufacturer’s problem amounts
to
max [π(q0 , 0) − Kq0 ] + [π(qk , k) − Kqk ] − F0 − Fk
q0 ,qk
(B.11)
From the manufacturer’s objective function it becomes apparent that the
optimal quantity to offer the retailer with low marginal cost is q0III = q JS (0).
Likewise, the optimal quantity to offer the retailer with high marginal cost
is qkIII = q JS (k).
It remains to check whether the neglected constraints are satisfied. First,
(IC0 ) is satisfied as long as
F0 ≥ kq JS (k) + Fk ⇐⇒ q JS (k) ≤
F
.
k
(B.12)
Likewise, (ICk ) is satisfied as long as
Fk ≥ −kq JS (0) + F0 ⇐⇒ q JS (0) ≥
4
F
.
k
(B.13)
Case IV: (PC0 ), (PCk ), and (ICk ) bind.
If (PC0 ) and (PCk ) are binding, transfers are given by
t0 = π(q0 , 0) − F0
tk = π(qk , k) − Fk .
and
(B.14)
Inserting these transfers into the binding (ICk ) constraint pins down the
quantity optimally offered to the retailer with low marginal cost:
Fk = π(q0 , k) − π(q0 , 0) + F0 ⇒ q0IV =
F
.
k
(B.15)
Ignoring (IC0 ) for the moment, the manufacturer’s problem amounts to
max [π(q0IV , 0) − Kq0IV ] + [π(qk , k) − Kqk ] − F0 − Fk .
qk
(B.16)
Hence, the optimal quantity to offer the retailer with high marginal cost is
q0IV = q JS (k).
The monotonicity requirement (MON) is satisfied as long as
q JS (k) ≤
F
,
k
(B.17)
in which case also (IC0 ) is satisfied because (ICk ) binds.
Case V: (PC0 ) and (ICk ) bind.
If (PC0 ) and (ICk ) are binding, transfers are given by
t0 = π(q0 , 0) − F0
tk = π(qk , k) + kq0 − F0 .
and
(B.18)
Ignoring (IC0 ) and (PCk ) for the moment, the manufacturer’s problem amounts
to
max [π(q0 , 0) − (K − k)q0 ] + [π(qk , k) − Kqk ] − 2F0
q0 ,qk
(B.19)
From the manufacturer’s objective function it becomes apparent that the
optimal quantity to offer the retailer with high marginal cost is qkV = q JS (k).
Differentiation w.r.t. q0 reveals that the optimal quantity to offer the retailer
with low marginal cost is q0V > 0, which is implicitly characterized by
P (q0V ) + q0V P 0 (q0V ) = K − k.
(B.20)
It remains to check whether the neglected constraints are satisfied. The
left-hand side of (B.20) is decreasing in q, such that q0V > q JS (0). This,
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in turn, implies that the monotonicity requirement (MON) is satisfied. As
(ICk ) binds, (IC0 ) then is automatically satisfied. Finally, (PCk ) is satisfied
as long as
Fk ≤ −kq0V + F0 ⇐⇒ q0V ≤
F
.
k
(B.21)
Noting that the manufacturer will always be (weakly) better off in a situation where only two (Cases I, III and V) rather than three constraints (Cases
II and IV) impose a binding restriction, the following result summarizes the
optimal quantities under uniform pricing.
Lemma 4. Let qiu denote the optimal quantity offered to downstream firm
i ∈ {0, k} under uniform pricing. Then:
(i) If
F
k
≤ qkI , then qku = qkI and q0u = q JS (0).
(ii) If qkI <
F
k
< q JS (k), then qku =
F
k
and q0u = q JS (0).
(iii) If q JS (k) ≤
F
k
≤ q JS (0), then qku = q JS (k) and q0u = q JS (0).
(iv) If q JS (0) <
F
k
< q0V , then qku = q JS (k) and q0u =
(v) If q0V ≤
F
,
k
F
.
k
then qku = q JS (k) and q0u = q0V .
Free Disposal.—Lemmas 3 and 4 characterize the optimal quantities for
the case of quantity forcing. Now suppose that downtream firms can freely
dispose of the input. Given downstream firm i ∈ {0, k} obtains quantity q̃ of
the input for free, it will sell min{q̃, q(ki )} units of the final product, where
q(ki ) is defined in (1) and satisfies P (q(ki ))+q(ki )P 0 (q(ki )) = ki . With q JS (k)
and q0V being defined by P (q JS (k)) + q JS (k)P 0 (q JS (k)) = K + k > k and
P (q0V ) + q0V P 0 (q0V ) = K − k > 0, respectively, and P (q) + qP 0 (q) bing strictly
decreasing, it follows that qku ≤ qkd = q JS < q(k) and q0d ≤ q0u ≤ q0V < q(0). In
consequence, free disposal leaves the quantities sold by each downstream firm
under each pricing regime unaffected. The quantities that the manufacturer
offers to the two downstream firms under the respective pricing regime, as
characterized in Lemmas 3 and 4, are depicted in Figure B.1
Welfare.—Social welfare does not depend on the specifics of the contractual form, but on the quantities of the final consumption good. As
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Figure B.1: Optimal quantities under general nonlinear contracts.
P (0) > K + k by assumption, each market should be served from a welfare perspective. The quantity that maximizes welfareR in the market served
q
by downstream firm i ∈ {0, k}, qiW , maximizes Wi = 0 P (z)dz − (K + ki )q
and thus is characterized by
P (qiW ) = K + ki .
(B.22)
Regarding the market served by the downstream firm with marginal cost k,
as P (qkW ) = K + k < K + k − q JS (k)P 0 (q JS (k)) = P (q JS (k)) and P 0 < 0
(whenever P > 0), we must have q JS (k) < qkW . Regarding the market
served by the downstream firm with marginal cost 0, note that we must have
tV0 − Kq0V ≥ 0, otherwise the manufacturer would be better off in Case V by
offering only the quantity-transfer pair (qkJS , tJS
k ), which would be rejected
by the downstream firm with marginal cost 0. With tV0 = q0V P (q0V ) − F0 , this
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implies q0V [P (q0V ) − K] ≥ F0 . As, in Case V, F0 > Fk ≥ 0 , we must have
JS
W
0
P (q0V ) > K = P (W
0 ), such that q (0) < qk as P < 0 (whenever P > 0).
With Wi being strictly concave in q (whenever P > 0), welfare in market
i ∈ {0, k} is higher under the pricing regime that leads to the larger quantity
being sold in that market. The welfare comparison across pricing regimes
stated in Proposition 4 then is immediate.
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